Higher-order MRFs based image super resolution: MMSE or MAP?

A trained filter-based higher-order Markov Random Fields (MRFs) model - the so called Fields of Experts (FoE), has proved a highly effective image prior model for many classic image restoration problems. Generally, two options are available to incorporate the learned FoE prior in the inference procedure: (1) sampling-based minimum mean square error (MMSE) estimate, and (2) energy minimization-based maximum a posteriori (MAP) estimate. It is well-known the sampling-based MMSE estimate is very time consuming, but the MAP inference has a remarkable advantage of high computational efficiency. In a recent paper, the FoE prior model was exploited for the single image super resolution (SR) task by using the MMSE inference, based on a seemingly correct conclusion that the MAP inference of the FoE prior based model, which leads a non-convex optimization problem, is prone to getting stuck in some bad local minima. However, in this letter, we demonstrate that this simpler inference criterion - the MAP estimate, works equally well compared to the complicated MMSE estimate with exactly the same prior model. Moreover, with our discriminatively trained FoE prior model, the MAP inference can even lead to further improvements. Consequently, we argue that for higher-order natural image prior based SR problem, it is not necessary to employ the time consuming MMSE estimation.